Asymptotics of supremum distribution of -locally stationary Gaussian processes

We study the exact asymptotics of P ( sup t ∈ [ 0 , S ] X ( t ) > u ) , as u → ∞ , for centered Gaussian processes with the covariance function satisfying 1 − C ov ( X ( t ) , X ( t + h ) ) = A ( t ) | h | α ( t ) + o ( | h | α ( t ) ) , as h → 0 . tained results complement those already considered in the literature for the case of locally stationary Gaussian processes in the sense of Berman, where α ( t ) ≡ α . It appears that the behavior of α ( t ) in the neighborhood of its global minimum on [ 0 , S ] significantly influences the asymptotics. illustration we work out the case of X ( t ) being a standardized multifractional Brownian motion.